One of the central problems in arithmetic statistics is counting number field extensions of a fixed degree with a given Galois group, ordered by discriminant. In this talk, we focus on extensions with Galois group of the form C2 ≀ H over an arbitrary base field. We begin by discussing the historical development of results in counting such extensions, including the work of Jürgen Klüners, who established the main term in this setting. We then turn to the problem of obtaining explicit power-saving error terms. Using Tauberian methods, we describe how such savings can be achieved, and present an alternative approach that leads to improved power-saving error terms in greater generality. We conclude with a brief discussion of possible directions for future work.